First Law Closed System Formulas

ISOMETRIC (Isochoric) PROCESS

a.) Relationship between P and T.

T1/T2 = P1/P2

b.) Non-flow work.

Wn =

c.) The change of internal energy

U = mcv(T2 T1)

d.) The heat transferred

Q = mcv(T2 T1)

e.) The change of enthalpy

H = mcp(T2 - T1)

f.) Irreversible non-flow constant volume process

Q = U + Wn

where: For reversible non-flow, Wn = 0

For irreversible nonflow, Wn 0

Wn = non-flow work

ISOBARIC PROCESS

a) Relationship between V and T

T1/T2 = V1/V2

b) Nonflow work

Wn = = P(V2 V1) = mR(T2 T1)

c) Change of internal energy

U = mcv (T2 T1)

d) Heat transferred

Q = mcp (T2 T1)

e) Change of enthalpy

H = mcp (T2 T1)

ISOTHERMAL PROCESS

(a.) Relationship between P and V

P1V1 = P2V2

(b.) Nonflow work

Wn = P1V1lnV2/V1 = mRT1lnV2/V1

(c.) The change of internal energy

U = O

(d.) The heat transferred

Q = Wn = P1V1lnV2/V1 = mRT1lnP1/P2

(e.) The change of enthalpy

H = 0

ISENTROPIC PROCESS

1. Relationship among P, V and T.

(a) Relationship between P and V

P1V1k = P2V2k = C

(b) Relation between T and V

From P1 V1k = P2 V2k and P1V1/T1 = P2V2/T2, we have

T2/T1 = [V1/V2]k-1

(c ) Relation between T and P

T2/T1 = [p2/p1](k-1)/k

2. Nonflow work.

From PVk = C, P = CV-k

Wn = PdV = C-kdV = C V-kdV

Integrating and simplifying

Wn = P2V2 P1V1 / (1- k) = mR(T2 T1) / (1 k)

3. The change of internal energy.

U = mCv (T2 T1)

4. The heat transferred.

Q = 0

5. The change of enthalpy.

H = mCp (T2 T1)

6. The change of entropy.

S = 0

POLYTROPIC PROCESS

1. Relationship among p, V, and T

(a) Relation between P and V

P1V1n = P2V2n

(b) Relation between T and V

T2/T1 = [V1/V2]n-1

(c) Relation between T and p

T2/T1 = [p2/p1](n-1)/n

2. Nonflow work

Wn = pdV = P2V2 P1V1 / 1-n = mR(T2-T1) / 1-n

3. The change of internal energy.

U = mCv(T2 T1)

4. The heat transferred. Q = U + Wn

= mcv (T2 T1) + mR (T2 T1)/(1 - n)

= m [(cv ncv + R)/(1 n)] (T2 T1)

= m [(cp ncv)/(1 n)] (T2 T1)

= mcv [(k n)/(1 n)] (T2 T1)

Q = mCn(T2 T1)

Cn = Cv [(k n)/(1 n)], the polytropic specific heat

5. The change of enthalpy

H = mCp (T2 T1)

First Law Open System Formulas

Mass:

Energy:

Nozzle/Diffuser:

2 2

1 21 20

2

V Vh h

Turbine and Compressor:

Heat Exchangers:

2 2

2 2

i ei i i e e e

i e

V Vm h gz m h gz

Throttling Device:

i eh h

Transient System

Mass:

Energy:

cvi e

i e

dmm m

dt

2 2

2 2

cv i ecv cv i i i e e e

i e

dE V VQ W m h gz m h gz

dt

2 2

1 21 2 1 2

2

W V Vh h g z z

m

0cv cv i ei e

m t m m m

0 0

0t t

cv cv cv cv i i e e

i e

U t U Q W m hdt m h dt